Table Of ContentFirst Edition, 2012
ISBN 978-81-323-0884-3
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Table of Contents
Chapter 1 - Fracture Mechanics
Chapter 2 - Fatigue (Material)
Chapter 3 - Fractography
Chapter 4 - Focal Mechanism
Chapter 5 - Fracture Toughness
Chapter 6 - Fracture
Chapter 7 - Structural Fracture Mechanics
Chapter 8 - Peridynamics
Chapter 9 - Bending
Chapter 10 - Euler–Bernoulli Beam Equation
Chapter 11 - Stress (Mechanics)
Chapter 12 - Shear Stress
Chapter 13 - Timoshenko Beam Theory
Chapter 1
Fracture Mechanics
Fracture mechanics is the field of mechanics concerned with the study of the
propagation of cracks in materials. It uses methods of analytical solid mechanics to
calculate the driving force on a crack and those of experimental solid mechanics to
characterize the material's resistance to fracture.
In modern materials science, fracture mechanics is an important tool in improving the
mechanical performance of materials and components. It applies the physics of stress and
strain, in particular the theories of elasticity and plasticity, to the microscopic
crystallographic defects found in real materials in order to predict the macroscopic
mechanical failure of bodies. Fractography is widely used with fracture mechanics to
understand the causes of failures and also verify the theoretical failure predictions with
real life failures.
An edge crack (flaw) of length a in a material.
Linear elastic fracture mechanics
Griffith's criterion
Fracture mechanics was developed during World War I by English aeronautical engineer,
A. A. Griffith, to explain the failure of brittle materials. Griffith's work was motivated by
two contradictory facts:
• The stress needed to fracture bulk glass is around 100 MPa (15,000 psi).
• The theoretical stress needed for breaking atomic bonds is approximately
10,000 MPa (1,500,000 psi).
A theory was needed to reconcile these conflicting observations. Also, experiments on
glass fibers that Griffith himself conducted suggested that the fracture stress increases as
the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used
extensively to predict material failure before Griffith, could not be a specimen-
independent material property. Griffith suggested that the low fracture strength observed
in experiments, as well as the size-dependence of strength, was due to the presence of
microscopic flaws in the bulk material.
To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental
specimens. The artificial flaw was in the form of a surface crack which was much larger
than other flaws in a specimen. The experiments showed that the product of the square
root of the flaw length (a) and the stress at fracture (σ) was nearly constant, which is
f
expressed by the equation:
An explanation of this relation in terms of linear elasticity theory is problematic. Linear
elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a
linear elastic material is infinite. To avoid that problem, Griffith developed a
thermodynamic approach to explain the relation that he observed.
The growth of a crack requires the creation of two new surfaces and hence an increase in
the surface energy. Griffith found an expression for the constant C in terms of the surface
energy of the crack by solving the elasticity problem of a finite crack in an elastic plate.
Briefly, the approach was:
• Compute the potential energy stored in a perfect specimen under an uniaxial
tensile load.
• Fix the boundary so that the applied load does no work and then introduce a crack
into the specimen. The crack relaxes the stress and hence reduces the elastic
energy near the crack faces. On the other hand, the crack increases the total
surface energy of the specimen.
• Compute the change in the free energy (surface energy − elastic energy) as a
function of the crack length. Failure occurs when the free energy attains a peak
value at a critical crack length, beyond which the free energy decreases by
increasing the crack length, i.e. by causing fracture. Using this procedure, Griffith
found that
where E is the Young's modulus of the material and γ is the surface energy density of the
material. Assuming E = 62 GPa and γ = 1 J/m2 gives excellent agreement of Griffith's
predicted fracture stress with experimental results for glass.
Irwin's modification
The plastic zone around a crack tip in a ductile material.
Griffith's work was largely ignored by the engineering community until the early 1950s.
The reasons for this appear to be (a) in the actual structural materials the level of energy
needed to cause fracture is orders of magnitude higher than the corresponding surface
energy, and (b) in structural materials there are always some inelastic deformations
around the crack front that would make the assumption of linear elastic medium with
infinite stresses at the crack tip highly unrealistic. F. Erdogan (2000)
Griffith's theory provides excellent agreement with experimental data for brittle materials
such as glass. For ductile materials such as steel, though the relation still
holds, the surface energy (γ) predicted by Griffith's theory is usually unrealistically high.
A group working under G. R. Irwin at the U.S. Naval Research Laboratory (NRL) during
World War II realized that plasticity must play a significant role in the fracture of ductile
materials.
In ductile materials (and even in materials that appear to be brittle), a plastic zone
develops at the tip of the crack. As the applied load increases, the plastic zone increases
in size until the crack grows and the material behind the crack tip unloads. The plastic
loading and unloading cycle near the crack tip leads to the dissipation of energy as heat.
Hence, a dissipative term has to be added to the energy balance relation devised by
Griffith for brittle materials. In physical terms, additional energy is needed for crack
growth in ductile materials when compared to brittle materials.
Irwin's strategy was to partition the energy into two parts:
• the stored elastic strain energy which is released as a crack grows. This is the
thermodynamic driving force for fracture.
• the dissipated energy which includes plastic dissipation and the surface energy
(and any other dissipative forces that may be at work). The dissipated energy
provides the thermodynamic resistance to fracture. Then the total energy
dissipated is
G = 2γ + G
p
where γ is the surface energy and G is the plastic dissipation (and dissipation from other
p
sources) per unit area of crack growth.
The modified version of Griffith's energy criterion can then be written as
For brittle materials such as glass, the su rface energy term dominates and
. For ductile materials such as steel, the plastic dissipation term
dominates and . For polymers close to the glass transition
temperature, we have intermediate values of .
Stress intensity factor
Another significant achievement of Irwin and his colleagues was to find a method of
calculating the amount of energy available for fracture in terms of the asymptotic stress
and displacement fields around a crack front in a linear elastic solid. This asymptotic
expression for the stress field around a crack tip is
where σ are the Cauchy stresses, r is the distance from the crack tip, θ is the angle with
ij
respect to the plane of the crack, and f are functions that are independent of the crack
ij
geometry and loading conditions. Irwin called the quantity K the stress intensity factor.
Since the quantity f is dimensionless, the stress intensity factor can be expressed in units
ij
of .
When a rigid line inclusion is considered, a similar asymptotic expression for the stress
fields is obtained.
Strain energy release
Irwin was the first to observe that if the size of the plastic zone around a crack is small
compared to the size of the crack, the energy required to grow the crack will not be
critically dependent on the state of stress at the crack tip. In other words, a purely elastic
solution may be used to calculate the amount of energy available for fracture.
The energy release rate for crack growth or strain energy release rate may then be
calculated as the change in elastic strain energy per unit area of crack growth, i.e.,
where U is the elastic energy of the system and a is the crack length. Either the load P or
the displacement u can be kept fixed while evaluating the above expressions.
Irwin showed that for a mode I crack (opening mode) the strain energy release rate and
the stress intensity factor are related by:
where E is the Young's modulus, ν is Poisson's ratio, and K is the stress intensity factor
I
in mode I. Irwin also showed that the strain energy release rate of a planar crack in a
linear elastic body can be expressed in terms of the mode I, mode II (sliding mode), and
mode III (tearing mode) stress intensity factors for the most general loading conditions.
Next, Irwin adopted the additional assumption that the size and shape of the energy
dissipation zone remains approximately constant during brittle fracture. This assumption
suggests that the energy needed to create a unit fracture surface is a constant that depends
only on the material. This new material property was given the name fracture toughness
and designated G . Today, it is the critical stress intensity factor K which is accepted as
Ic Ic
the defining property in linear elastic fracture mechanics.
Limitations
The S.S. Schenectady split apart by brittle fracture while in harbor (1944)
But a problem arose for the NRL researchers because naval materials, e.g., ship-plate
steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a
crack. One basic assumption in Irwin's linear elastic fracture mechanics is that the size of
the plastic zone is small compared to the crack length. However, this assumption is quite
restrictive for certain types of failure in structural steels though such steels can be prone
to brittle fracture, which has led to a number of catastrophic failures.
Linear-elastic fracture mechanics is of limited practical use for structural steels for
another more practical reason. Fracture toughness testing is very expensive and engineers
believe that sufficient information for selection of steels can be obtained from the simpler
and cheaper Charpy impact test.
Nonlinear elasticity and plasticity
Vertical stabilizer, which separated from American Airlines Flight 587, leading to a fatal
crash
Most engineering materials show some nonlinear elastic and inelastic behavior under
operating conditions that involve large loads. In such materials the assumptions of linear
elastic fracture mechanics may not hold, that is,
• the plastic zone at a crack tip may have a size of the same order of magnitude as
the crack size
• the size and shape of the plastic zone may change as the applied load is increased
and also as the crack length increases.
Therefore a more general theory of crack growth is needed for elastic-plastic materials
that can account for:
• the local conditions for initial crack growth which include the nucleation, growth,
and coalescence of voids or decohesion at a crack tip.
• a global energy balance criterion for further crack growth and unstable fracture.
R-curve
An early attempt in the direction of elastic-plastic fracture mechanics was Irwin's crack
extension resistance curve or R-curve. This curve acknowledges the fact that the
resistance to fracture increases with growing crack size in elastic-plastic materials. The
